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A Centrality Measure for Electrical Networks

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A Centrality Measure for Electrical Networks Carnegie Mellon Electricity Industry Center Working Paper CEIC-07 www.cmu.edu/electricity 1 A Centrality Measure for Electrical Networks Paul Hines and Seth Blumsack types of failures. Many classifications of network structures Abstract—We derive a measure of “electrical centrality” for have been studied in the field of complex systems, statistical AC power networks, which describes the structure of the mechanics, and social networking [5,6], as shown in Figure 2, network as a function of its electrical topology rather than its but the two most fruitful and relevant have been the random physical topology. We compare our centrality measure to network model of Erdös and Renyi [7] and the “small world” conventional measures of network structure using the IEEE 300- bus network. We find that when measured electrically, power model inspired by the analyses in [8] and [9]. In the random networks appear to have a scale-free network structure. Thus, network model, nodes and edges are connected randomly. The unlike previous studies of the structure of power grids, we find small-world network is defined largely by relatively short that power networks have a number of highly-connected “hub” average path lengths between node pairs, even for very large buses. This result, and the structure of power networks in networks. One particularly important class of small-world general, is likely to have important implications for the reliability networks is the so-called “scale-free” network [10, 11], which and security of power networks. is characterized by a more heterogeneous connectivity. In a Index Terms—Scale-Free Networks, Connectivity, Cascading scale-free network, most nodes are connected to only a few Failures, Network Structure others, but a few nodes (known as hubs) are highly connected to the rest of the network. The signature property of a scale- I. INTRODUCTION free network is a power-law distribution (a very fat-tailed N bulk electric power networks, wide-scale cascading distribution) of node connectivity or degree. I failures happen more often than would be expected if failures were random and independent, and their sizes followed a normal distribution [1]. Despite reliability standards and significant technological advances, the frequency of large blackouts in the United States has not decreased since the creation of NERC in 1965 [2]. The US blackout of August 2003, along with the Italian blackout of September 2003, convinced even skeptics that fundamental weaknesses exist in transmission infrastructures. Many have sought to explain this weakness as a function of a shift in electricity industry structure from regulated, vertically integrated utilities with moderate interregional trade, to a diverse set of market participants using the transmission infrastructure to facilitate long distance energy transactions [3,4]. While the merits and shortcomings of electricity market restructuring are a contentious subject, at the very least it is fair to say that the number and size of blackouts on the North American electric grid has not decreased in recent years, as Figure 1: The number of large blackouts according to [NERC DAWG] shown in Figure 1. between 1984 and 2006. The bars show the data after adjusting for demand growth (using year 2000 as the base year). The line shows the number of Recent advances in network and graph theory have drawn blackouts >400 MW before this adjustment. Clearly, the frequency of large links between the topological structure of networks cascading failures in the North American electric power system has not (particularly networks consisting of social ties and decreased during the 23 year period shown here. Assuming that large failures infrastructures) and the vulnerability of networks to certain were not more common before this period, the overall frequency of large blackouts has not decreased since the formation of NERC in 1965. This work was supported in part by the Alfred P. Sloan Foundation and the Electrical Power Research Institute through grants to the Carnegie Mellon The structural differences between the random and scale- Electricity Industry Center and in part by the US Dept. of Energy through free network models have important implications for network grants to the National Energy Technology Laboratory. vulnerability. In particular, random networks tend to be fairly Paul Hines is with the School of Engineering, University of Vermont, Burlington, VT 05405 (e-mail: [email protected]). robust to targeted attacks but relatively vulnerable to a series Seth Blumsack is with the Dept. of Energy and Mineral Engineering, of random failures or attacks. Scale-free networks, on the Pennsylvania State University, University Park, PA 16902 (e-mail: other hand, are highly vulnerable to targeted attacks or failures [email protected]). at one of the hubs, but are more robust to failures at randomly Carnegie Mellon Electricity Industry Center Working Paper CEIC-07 www.cmu.edu/electricity 2 chosen nodes [34]. Similarly, a network’s structure has Section VI offers some conclusions and extensions. important implications for strategies aimed at protecting the network against cascading failures or more deliberate attacks.2 II. NETWORK CONNECTIVITY METRICS AND APPLICATIONS TO The world wide web is a good example of a scale-free POWER NETWORKS network; the network as a whole can be made more robust by Network connectivity is often measured using the degree increasing the reliability of a fairly small set of critical hubs of nodes in the network – the number of edges (and thereby (such as google.com or yahoo.com). other nodes) connected to a given node. The degree, k, distribution varies substantially from one network structure to another. In a regular lattice structure (such as the nearest neighbor and second neighbor graphs shown in Figure 2) k is constant for all nodes. Random networks have an exponential degree distribution, as do most small-world networks. When a small-world network shows a power-law degree distribution, rather than an exponential, it is known as a scale-free network [11]. If P(k) is the probability that a randomly chosen node has degree k, the degree distribution of a scale free network follows Pk()~ k−γ . In most real networks, γ falls in the range of 2 to 3. Many seemingly different types of networks have been found to possess a scale-free structure, including the world wide web, airline networks, and protein interactions in yeast [5]. Power networks represent a natural test case for complex Figure 2: Small examples of the network structures used in this paper. systems and network theory. However, the lack of widely The principal shortcoming of existing attempts to measure available data on the actual topology of the power grid, the network structure of electrical grids lies in the failure to particularly in the United States, has dramatically limited the explicitly incorporate the physical laws governing the flow of number of analyses. The few existing studies generally find electricity within these networks. While the topological (node- that power networks exhibit some kind of nonrandom edge) structure of an electrical power network may suggest structure, but there is disagreement over the actual structure of one set of behaviors and vulnerabilities, the electrical structure the network. In particular the existing studies disagree with of the network may suggest something altogether different. respect to whether the tail of the degree distribution follows a The flow of power through the network is governed by power law or exponential distribution. Portions of the North Kirchoff’s Laws and not simply by topology. To address this American grid have been discussed in [5, 6, 16, 18 – 20]. The shortcoming we introduce an electrical connectivity metric North American grid as a whole is discussed in [18], in which which incorporates the information contained in the system the authors report that the degree distribution of the power impedance matrix. This metric is referred to here as “electrical grid has an exponential or “single-scale” [11] form with a fast- centrality.” Electrical centrality accommodates Kirchoff’s decaying tail, although the distribution of the number of Laws and is more accurately captures the properties of node transmission lines passing through a given node (the centrality, relative to metrics based on node-edge connectivity. “betweenness” of a given node) does follow a power law. The This paper is organized in five sections. Section I is this authors of [16] find the same single-scale structure in the introduction, and section II reviews the existing literature degree distribution for a portion of the electric network in related to the subject matter. Section III reviews the classical California. The same result appears in [5] and [6] for the network connectivity metrics, which provide information only power grid of the Western United States. However, [10, 17] on the topological structure of the network in question, and estimate a power-law relationship in the degree distribution discusses some existing applications of these metrics to power for both the Western [10, 17] and Eastern [17] portions of the networks. Section IV introduces the electrical centrality U.S. power grid. metric. In Section V we use the IEEE 300-bus network to Structural properties of the Italian, French, and Spanish illustrate the differences between our electrical connectivity power networks can be found in [19] and [20]. These metric and the classical topological connectivity metrics. European power networks are generally found to have the same single-scale topological structure as the North American 2 It is important to realize that, particularly in complex systems such as grid, with an exponential tail in the degree distribution. The power networks, building more is not always better. The North American distribution of nodal demands (or loads), however, follows a blackout of August 2003 was followed by a number of calls for massive investments in transmission infrastructure; [12], for example, suggested a power law: most buses in the grid serve lower-than average program of investment and grid modernization amounting to $100 billion.
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